The Physics of Euler's Formula - Laplace Transform Prelude

Students often visualize derivatives as slopes of graphs, but this represents only one perspective. A more dynamic approach treats derivatives as velocity vectors describing motion through number space.

When the constant in the exponent is imaginary number i, the chain rule tells us the derivative of e^(it) is i times the function itself. Geometrically, multiplying by i acts like 90-degree rotation—velocity always perpendicular to position.

To get one of the most famous equations in all of math, if you wait for π units of time while rotating around the unit circle at unit speed, you end up precisely halfway around the circle. This is why e^(πi) = -1.

Engineers refer to this as the S-plane, which essentially means you should think of each point on that plane as encoding the entire function e^(st). This provides compact representation of infinite family of exponential behaviors.

The mass-on-spring setup represents a central example used throughout physics, describing position x changing over time with derivative giving velocity and second derivative giving acceleration.

There’s one very bizarre trick which really bothered calculus students when first encountered: you simply guess that the answer looks like e^(st), where s is just some constant you’re going to solve for.

The easiest case for the mass-spring system ignores damping coefficient (setting μ = 0). With rearrangement and taking square root, you find s equals plus or minus the square root of -k/m.

To connect pure mathland answer to something actually physical, you need to squeeze out real-valued solution from complex result e^(iomegat).

A critical property of our equation determines solution flexibility: it’s what we call linear equation, meaning if you have two distinct functions that solve it, when you add those functions, that sum also solves the differential equation.

Think about what it means if we reintroduce damping coefficient μ, setting it to something not equal to zero. In this case, solving equation looks like applying quadratic formula.

You can think about these functions e^(st) as being like the atoms of calculus. Complicated functions describing our world can often be broken up into these parts.